Journal of High Energy Physics

, 2018:112 | Cite as

Page curves for general interacting systems

  • Hiroyuki Fujita
  • Yuya O. Nakagawa
  • Sho Sugiura
  • Masataka WatanabeEmail author
Open Access
Regular Article - Theoretical Physics


We calculate in detail the Renyi entanglement entropies of cTPQ states as a function of subsystem volume, filling the details of our prior work [24], where the formulas were first presented. Working in a limit of large total volume, we find universal formulas for the Renyi entanglement entropies in a region where the subsystem volume is comparable to that of the total system. The formulas are applicable to the infinite temperature limit as well as general interacting systems. For example we find that the second Renyi entropy of cTPQ states in terms of subsystem volume is written universally up to two constants, (S2() = − ln K(β) + ln a(β) − ln 1+a(β)L+2), where L is the total volume of the system and a and K are two undetermined constants. The uses of the formulas were already presented in our prior work and we mostly concentrate on the theoretical aspect of the formulas themselves. Aside from deriving the formulas for the Renyi Page curves, the expression for the von Neumann Page curve is also derived, which was not presented in our previous work.


Lattice Quantum Field Theory Random Systems Conformal Field Theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  3. [3]
    M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
  4. [4]
    S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].MathSciNetGoogle Scholar
  5. [5]
    T. Takayanagi and T. Ugajin, Measuring black hole formations by entanglement entropy via coarse-graining, JHEP 11 (2010) 054 [arXiv:1008.3439] [INSPIRE].CrossRefzbMATHGoogle Scholar
  6. [6]
    T. Takayanagi and T. Ugajin, Measuring black hole formations by entanglement entropy via coarse-graining, JHEP 11 (2010) 054 [arXiv:1008.3439] [INSPIRE].CrossRefzbMATHGoogle Scholar
  7. [7]
    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.Google Scholar
  8. [8]
    P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech. (2007) P06008.Google Scholar
  9. [9]
    J.R. Garrison and T. Grover, Does a single eigenstate encode the full hamiltonian?, Phys. Rev. X 8 (2018) 021026.CrossRefGoogle Scholar
  10. [10]
    A.M. Kaufman et al., Quantum thermalization through entanglement in an isolated many-body system, Science 353 (2016) 794.CrossRefGoogle Scholar
  11. [11]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys. 88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].CrossRefGoogle Scholar
  13. [13]
    M. Hotta and A. Sugita, The fall of the black hole firewall: natural nonmaximal entanglement for the page curve, Prog. Theor. Exp. Phys. 2015 (2015) 123B04.Google Scholar
  14. [14]
    D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
  15. [15]
    S. Sugiura and A. Shimizu, Thermal pure quantum states at finite temperature, Phys. Rev. Lett. 108 (2012) 240401 [arXiv:1112.0740] [INSPIRE].CrossRefGoogle Scholar
  16. [16]
    S. Sugiura and A. Shimizu, Canonical thermal pure quantum state, Phys. Rev. Lett. 111 (2013) 010401 [arXiv:1302.3138] [INSPIRE].CrossRefGoogle Scholar
  17. [17]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.CrossRefGoogle Scholar
  18. [18]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.Google Scholar
  19. [19]
    M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature. 452 (2008) 854.CrossRefGoogle Scholar
  20. [20]
    A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83 (2011) 863 [arXiv:1007.5331] [INSPIRE].CrossRefGoogle Scholar
  21. [21]
    G. Biroli, C. Kollath and A.M. Läuchli, Effect of rare fluctuations on the thermalization of isolated quantum systems, Phys. Rev. Lett. 105 (2010) 250401.CrossRefGoogle Scholar
  22. [22]
    E. Iyoda, K. Kaneko and T. Sagawa, Fluctuation theorem for many-body pure quantum states, Phys. Rev. Lett. 119 (2017) 100601.MathSciNetCrossRefGoogle Scholar
  23. [23]
    A. Dymarsky, N. Lashkari and H. Liu, Subsystem eigenstate thermalization hypothesis, Phys. Rev. E 97 (2018) 012140 [arXiv:1611.08764] [INSPIRE].Google Scholar
  24. [24]
    Y.O. Nakagawa, M. Watanabe, S. Sugiura and H. Fujita, Universality in volume-law entanglement of scrambled pure quantum states, Nature Commun. 9 (2018) 1635 [arXiv:1703.02993] [INSPIRE].CrossRefGoogle Scholar
  25. [25]
    T.-C. Lu and T. Grover, Renyi entropy of chaotic eigenstates, arXiv:1709.08784 [INSPIRE].
  26. [26]
    Y. Huang, Universal eigenstate entanglement of chaotic local hamiltonians, to be published Nucl. Phys. (2018).Google Scholar
  27. [27]
    T. Faulkner, R.G. Leigh and O. Parrikar, Shape dependence of entanglement entropy in conformal field theories, JHEP 04 (2016) 088 [arXiv:1511.05179] [INSPIRE].MathSciNetGoogle Scholar
  28. [28]
    A. Sugita and A. Shimizu, Correlations of observables in chaotic states of macroscopic quantum systems, J. Phys. Soc. Japan 74 (2005) 1883.CrossRefzbMATHGoogle Scholar
  29. [29]
    I. Dumitriu, Eigenvalue statistics for beta-ensembles, technical report (2003).Google Scholar
  30. [30]
    S.A. Blanco and T.K. Petersen, Counting Dyck paths by area and rank, arXiv:1206.0803.
  31. [31]
    P. Reimann, Foundation of statistical mechanics under experimentally realistic conditions, Phys. Rev. Lett. 101 (2008) 190403 [arXiv:0810.3092] [INSPIRE].CrossRefGoogle Scholar
  32. [32]
    H.F. Trotter, On the product of semi-groups of operators, Proc. Math. Math. Soc. 10 (1959) 545.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    M. Suzuki, Generalized Trotters formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Comm. Math. Phys. 51 (1976) 183.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    N. Ullah, Invariance hypothesis and higher correlations of hamiltonian matrix elements, Nucl. Phys. 58 (1964) 65.MathSciNetCrossRefGoogle Scholar
  35. [35]
    T. Mansour and Y. Sun, Identities involving Narayana polynomials and Catalan numbers, arXiv:0805.1274.
  36. [36]
    R. Szmytkowski, On the derivative of the legendre function of the first kind with respect to its degree, J. Phys. A 39 (2006) 15147.MathSciNetzbMATHGoogle Scholar
  37. [37]
    F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST handbook of mathematical functions, Cambridge University Press, Cambridge U.K. (2010).zbMATHGoogle Scholar
  38. [38]
    N.M. Temme, Asymptotic methods for integrals, Series in Analysis volume 6, World Scientific, Hackensack U.S.A. (2015).Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute for Solid State PhysicsThe University of TokyoKashiwaJapan
  2. 2.Department of Physics, Faculty of ScienceThe University of TokyoTokyoJapan
  3. 3.Department of PhysicsHarvard UniversityCambridgeU.S.A.
  4. 4.Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced StudyThe University of TokyoKashiwaJapan

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